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University of Kansas

1. Se, Tony. Depth and Associated Primes of Modules over a Ring.

Degree: PhD, Mathematics, 2016, University of Kansas

URL: http://hdl.handle.net/1808/21895

This thesis consists of three main topics. In the first topic, we let R be a commutative Noetherian ring, I,J ideals of R, M a finitely generated R-module and F an R-linear covariant functor. We ask whether the sets \operatorname{Ass}_{R} F(M/I^{n} M) and the values \operatorname{depth}_{J} F(M/I^{n} M) become independent of n for large n. In the second topic, we consider rings of the form R = k[x^{a},x^{p1}y^{q1}, …,x^{pt}y^{qt},y^{b}], where k is a field and x,y are indeterminates over k. We will try to formulate simple criteria to determine whether or not R is Cohen-Macaulay. Finally, in the third topic we introduce and study basic properties of two types of modules over a commutative Noetherian ring R of positive prime characteristic. The first is the category of modules of finite F-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call F-abundant modules. These include, for example, the ring R itself and the canonical module when R has positive splitting dimension. We prove many facts about these two categories and how they are related. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest, such as complete intersections and invariant subrings.
*Advisors/Committee Members: Dao, Hailong (advisor), Jiang, Yunfeng (cmtemember), Katz, Daniel (cmtemember), Lang, Jeffrey (cmtemember), Nutting, Eileen (cmtemember).*

Subjects/Keywords: Mathematics; Cohen-Macaulay; coherent functors; divisor class group; F -regularity; Frobenius endomorphism; semigroup rings

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APA (6^{th} Edition):

Se, T. (2016). Depth and Associated Primes of Modules over a Ring. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21895

Chicago Manual of Style (16^{th} Edition):

Se, Tony. “Depth and Associated Primes of Modules over a Ring.” 2016. Doctoral Dissertation, University of Kansas. Accessed October 28, 2020. http://hdl.handle.net/1808/21895.

MLA Handbook (7^{th} Edition):

Se, Tony. “Depth and Associated Primes of Modules over a Ring.” 2016. Web. 28 Oct 2020.

Vancouver:

Se T. Depth and Associated Primes of Modules over a Ring. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2020 Oct 28]. Available from: http://hdl.handle.net/1808/21895.

Council of Science Editors:

Se T. Depth and Associated Primes of Modules over a Ring. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/21895

University of Southern California

2.
Narayanan, Anand Kumar.
Computation of *class* groups and residue *class* rings of
function fields over finite fields.

Degree: PhD, Computer Science, 2014, University of Southern California

URL: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/451911/rec/1538

We study the computation of the structure of two
finite abelian groups associated with function fields over finite
fields: the degree zero divisor class group and the multiplicative
group of the finite field itself. In addition, we present a novel
algorithm to factor polynomials over finite fields using Carlitz
modules. ❧ Let F_{q} denote the finite field with q elements and let
k = F_{q}(t) be the rational function field with constant field F_{q}.
Let F/k be a finite geometric abelian extension where an
F_{q}‐rational place in k denoted as ∞ splits completely. Stark units
are certain functions in F whose existence is claimed in the
function field analogue of the Brumer‐Stark conjecture. In the
function field setting, the Brumer‐Stark conjecture and hence the
existence of Stark units was proven by P. Deligne. Further, Stark
units are determined by the evaluations at 0 of Artin's L‐functions
associated with the complex characters of Gal(F/k). We prove that
for all primes ℓ not dividing q[F : k], the structure of the ℓ‐part
of the divisor class group of F is determined by Kolyvagin
derivative classes that are constructed out of Euler systems
associated with the Stark units. ❧ Further, given a certain
ℤ[Gal(F/k)] generator of the Stark units, we describe an algorithm
to compute the structure of the ℓ‐part of the divisor class group.
When F/k is a narrow ray class field (or a small index subextension
of a narrow ray class field), such a generator of the Stark units
module can be efficiently computed. ❧ The divisor class group
Cl^{0}_{F} of F is a finite abelian group and fits in the exact
sequence ❧ 0 → R_{F} → Cl^{0}_{F} → pic(O_{F}) → 0 ❧ where R_{F} is the
regulator and pic(O_{F}) is the ideal class group of O_{F}. Our
algorithm to compute the ℓ‐part of the divisor class group is
heavily reliant on the machinery of Euler systems of Stark units
and is efficient if the ℓ‐part of the ideal class group is small.
Empirical and heuristic evidence point to the ideal class group
being of very small order in comparison to the divisor class group.
❧ Other applications of our technique include a fast algorithm for
computing the divisor class number of narrow ray class extensions.
❧ We next turn to computing primitive elements in finite fields of
small characteristic. The multiplicative group of a finite field is
cyclic and generators (primitive elements) are abundant. However,
finding one efficiently remains an unsolved problem. We describe a
deterministic algorithm for finding a generating element of the
multiplicative group of the finite field F_{pn} with pⁿ elements
where p is a prime. In time polynomial in p and n, the algorithm
either outputs an element that is provably a generator or declares
that it has failed in finding one. Under a heuristic assumption,
the algorithm does succeed in finding a generator. The algorithm
relies on a relation generation technique in a recent breakthrough
by Antoine Joux's for discrete logarithm computation in small
characteristic finite fields. ❧ Building upon Joux's algorithm,
Barbulescu, Gaudry, Joux and Thome…
*Advisors/Committee Members: Huang, Ming-Deh (Committee Chair), Adleman, Leonard (Committee Member), Kamienny, Sheldon (Committee Member).*

Subjects/Keywords: number theory; computational number theory; function fields; finite fields; primitive elements; polynomial factorization; divisor class group; Stark units

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Narayanan, A. K. (2014). Computation of class groups and residue class rings of function fields over finite fields. (Doctoral Dissertation). University of Southern California. Retrieved from http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/451911/rec/1538

Chicago Manual of Style (16^{th} Edition):

Narayanan, Anand Kumar. “Computation of class groups and residue class rings of function fields over finite fields.” 2014. Doctoral Dissertation, University of Southern California. Accessed October 28, 2020. http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/451911/rec/1538.

MLA Handbook (7^{th} Edition):

Narayanan, Anand Kumar. “Computation of class groups and residue class rings of function fields over finite fields.” 2014. Web. 28 Oct 2020.

Vancouver:

Narayanan AK. Computation of class groups and residue class rings of function fields over finite fields. [Internet] [Doctoral dissertation]. University of Southern California; 2014. [cited 2020 Oct 28]. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/451911/rec/1538.

Council of Science Editors:

Narayanan AK. Computation of class groups and residue class rings of function fields over finite fields. [Doctoral Dissertation]. University of Southern California; 2014. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/451911/rec/1538

University of Tennessee – Knoxville

3.
Lynch, Benjamin Ryan.
Elasticity of Krull Domains with Infinite *class*="hilite">Divisor *Class* * Group*.

Degree: 2010, University of Tennessee – Knoxville

URL: https://trace.tennessee.edu/utk_graddiss/821

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has been determined which distributions of the height-one prime ideals will make R a half-factorial domain (HFD). For the group Q, certain distributions of height-one prime ideals are proven to make R an HFD. Finally, the last chapter studies general infinite groups and groups involving direct sums with Z. If certain conditions are met, then the elasticity of these divisor class groups is the same as the elasticity of simpler divisor class groups.

Subjects/Keywords: Krull domains; divisor class group; elasticity; half-factorial domain; block monoid; non-unique factorization; Algebra

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Lynch, B. R. (2010). Elasticity of Krull Domains with Infinite Divisor Class Group. (Doctoral Dissertation). University of Tennessee – Knoxville. Retrieved from https://trace.tennessee.edu/utk_graddiss/821

Chicago Manual of Style (16^{th} Edition):

Lynch, Benjamin Ryan. “Elasticity of Krull Domains with Infinite Divisor Class Group.” 2010. Doctoral Dissertation, University of Tennessee – Knoxville. Accessed October 28, 2020. https://trace.tennessee.edu/utk_graddiss/821.

MLA Handbook (7^{th} Edition):

Lynch, Benjamin Ryan. “Elasticity of Krull Domains with Infinite Divisor Class Group.” 2010. Web. 28 Oct 2020.

Vancouver:

Lynch BR. Elasticity of Krull Domains with Infinite Divisor Class Group. [Internet] [Doctoral dissertation]. University of Tennessee – Knoxville; 2010. [cited 2020 Oct 28]. Available from: https://trace.tennessee.edu/utk_graddiss/821.

Council of Science Editors:

Lynch BR. Elasticity of Krull Domains with Infinite Divisor Class Group. [Doctoral Dissertation]. University of Tennessee – Knoxville; 2010. Available from: https://trace.tennessee.edu/utk_graddiss/821